Questions and Answers
If a linear system has four equations and seven variables, then it must have infinitely many solutions.
False
If a matrix is in reduced row echelon form, then the first nonzero entry in each row is a 1 and has 0s below it.
True
If the first and second rows of an augmented matrix are (1,1,0) and (0,1,0) respectively, then the matrix is not in reduced row echelon form.
True
If the number of rows of an augmented matrix in reduced row echelon form is greater than the number of columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.
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Span{a1,a2} contains only the line through a1 and the origin, and the line through the a2 and the origin.
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There are exactly three vectors in span {a1,a2,a3}.
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The solution set of the linear system whose augmented matrix [a1a2a3b] is the same as the solution set of the equation x1a1+x2a2+a3x3=b.
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Give an example of a matrix A such that (1) Ax=b has a solution for infinitely many b∈R3, but (2) Ax=b does not have a solution for all b∈R3.
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A homogeneous system is always consistent.
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There is a vector [b1b2] so that the solutions to; 1 0 1 times x1 = b1 = z axis 0 1 0 x2 b2 x3.
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The solution set of a consistent inhomogeneous system Ax=b is obtained by translating the solution set of Ax=0.
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The equation Ax=b is homogeneous if the zero vector is a solution.
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The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.
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If x is a nontrivial solution of Ax=0, then every entry of x is nonzero.
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Give an example of a matrix A and a vector b such that the solution set of Ax=b is a line in R3 that does not contain the origin.
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If S is a set of linearly dependent vectors, then every vector in S can be written as a linear combination of the other vectors in S.
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The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution.
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If a set S of vectors contains fewer vectors than there are entries in the vectors, then the set must be linearly independent.
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Study Notes
Linear Algebra True/False Concepts
- Linear systems with more variables than equations can still be inconsistent, leading to no solutions.
- Reduced row echelon form requires the first nonzero entry in each row to be a 1, with zeroes beneath it.
- Specific row structures in an augmented matrix can indicate whether it is in reduced row echelon form.
- An augmented matrix with more rows than columns does not guarantee infinitely many solutions.
- The span of a set of vectors includes all possible linear combinations, not just the lines through individual vectors.
- A span can consist of infinitely many vectors, not just limited quantities.
- The solution sets for the augmented matrix and the corresponding equation will be equivalent.
- Certain matrix structures allow solutions for infinitely many outputs while failing for others.
- Homogeneous systems always have the trivial solution (the zero vector), making them consistent.
- An equation can fail to have a vector solution that fits specific linear combinations.
- Inconsistent systems can be resolved by translating the solution set from a corresponding homogeneous system.
- A homogeneous equation is defined by having the zero vector as a solution.
- The trivial solution for a homogeneous system exists regardless of free variables' presence.
- A nontrivial solution can have only one nonzero entry among several zero entries.
- Specific matrix and vector combinations can yield line solutions in R3 that do not include the origin.
- A set of linearly dependent vectors does not require all vectors to be expressible as combinations of others, only one.
- The trivial solution for Ax=0 exists for all systems, detached from the linear independence of columns.
- A set with fewer vectors than their dimensions doesn't automatically imply linear independence; counterexamples exist.
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Description
Test your knowledge of linear algebra concepts with this true/false quiz. Each statement challenges your understanding of linear systems, matrices, and their properties. Perfect for students looking to solidify their grasp of the subject.
